I am reading "Investigations" by Stuart Kauffman. It seems very much related to some of my own pet obsessions, despite primarily being concerned with biology.

Imagine a probability density function that is non-zero within a helix but zero outside. Also the probability density is low and one end and high at the other. A Metropolis-Hastings random walk is initialized with a sample at the low density end.

What happens?

The sample moves along the spiral.

It moves! A reversible Markov chain system and yet it moves as though there is time! Ain't that just so cool!

[Update: Alternately, the Metropolis-Hastings algorithm may be set up to use a catalytic step selection as per my previous entry. In this case, an explicit helix does not need to be specified for the density function, the helical motion is an emergent behaviour.]

Kauffman's thesis (if i'm reading him correctly) is that such a spiral represents the basis of life. The movement down the helix represents use of energy, the spiral a resultant guided action. More complex actions can of course also be imagined. Non-living systems fall straight to high density regions, but life approaches it by strange pathways.

Everything seems to follow this pattern. This is a description of the universe, and of the human mind. I'm willing to bet i can make a tile-assembler that displays this behaviour too, though tiles themselves are timeless.